Hi, I'm Vladimir Podolski and today we're going to discuss divisibility. So we are used to numbers. We use them constantly every day, everywhere in our life. But, what is the reason that we have our numbers? Why do we have them to satisfy our needs? How it happened that we have specifically numbers we're having. So let's review the history. It all starts with counting. So, the most basic task is to count some objects and so we have numbers one, two, three and so on. Okay, and already now, we have simple basic operations, we have addition and multiplication, they are possible. So and these are basic tasks. So if you have for example, two flocks and we unite them then we need addition to count the number of sheep in the resulting flock. So addition and multiplication are possible and okay, but what about in their inverse operations? Are they possible? So can we reverse the operations we're having? Can we reverse addition and multiplication. Okay. And, an inverse to addition is subtraction. So suppose we have added four to five. So the result is nine. So now to obtain five back, we have to subtract four. So, subtraction is an inverse operation to addition. And in the same way, division is an inverse operation to multiplication. So suppose we have multiplied three by four and the result is 12. So now, to get back our original number, we have to divide the result by four and the result will be three. So, are these inverse operations possible for our numbers that we currently have one, two, three and so on, numbers we're having to count? And it turns out that already subtraction is not always possible. For example, we cannot subtract three from two and obtain some positive integer. If you have a group of two objects, we cannot not take three objects out of them. Okay, so we have a problem. We would like to have subtraction and it is not always possible. So what is the solution? And the solution is very familiar and well known to us. We have to introduce negative numbers and zero. Note that these ideas look simple, so to add negative numbers and zero. But actually, these are deep ideas and they took humanity a lot of time to invent them. And these are actually two separate deep ideas. So now, we have numbers one, two, three and so on to count. Now we have added number zero and we also have added negative numbers minus one, minus two and so on. And now, subtraction is always possible. So here are some examples, two minus three is minus one. And here are other examples, so the rules are well known. And also, our original operations, we already had addition and multiplication, can be extended to negative numbers as well. So here are some examples as well, but again, the rules are well known. Now, we have addition, subtraction, and multiplication, and we have integer numbers, positive and negative. And so, but what about division? And it turns out that division is not always possible. For example, we cannot divide three by two and obtain some integer number. This is not possible. Again, so what is the solution and the idea standard for us. But again, it's a deep idea, to introduce rational numbers. We would consider fractions and then three over two makes perfect sense. But sometimes, it is important to stay with integers. Sometimes, it is important to have an integer answer. Some things are not divisible and we cannot consider fractions for them. And so, we have stayed within integer numbers and not within rational numbers. So we have to live with the situation that division is not always possible. And so, we arrive at Number Theory. Number Theory studies integers and operations on them. And of course, basics of Number Theory have natural applications. So, this is a very natural branch of mathematics, we study integer numbers and standard operations on them. So we have natural real life applications. But it turned out that deep results in Number Theory had no applications. So, if you can see they are more advanced, more involved results regarding integers and operations on them, then they have no applications. And this was stated explicitly and well understood by top number theorists including Hardy and Dickson, and they actually praise Number Theory for this. They talked about Number Theory as a branch of Pure Mathematics, Mathematics for itself, and so on. Okay. Does it mean that the Number Theory is useless? Does it mean that we don't have to study it? Of course, we have to study the basics of Number Theory to have simple applications, but probably the rest is not needed at all. And it turns out that it's not. Not anymore. Things changed dramatically in computer era. And here is the quote by Donald Knuth, the famous computer scientist. It turns out that results in Number Theory are important for high speed numerical calculations and so they're crucial for Computer Science. But this is only a part of a story. And another big part of this story is that, Number Theory is vital for modern cryptography. And through cryptography, Number Theory dramatically affects our life. It is used in emails, in organization of messengers, in online transactions, and in the internet as a whole, and so on. So it affects many size of our life and so it is very important.